Math  /  Data & Statistics

Question7. A Gallup poll asked random samples of Americans in 2016 and 2018 if they were satisfied with the quality of the environment. In 2016, 548 were satisfied with the quality of the environment and 430 were dissatisfied. In 2018, 477 were satisfied and 438 were dissatisfied. Determine whether the proportion of Americans who are satisfied with the quality of the environment has declined. Use a 0.05 significance level. a) Will you conduct hypotheses tests for mean or proportion? One sample or two sample? Explain. Utilize the correct variables to support your answers. conduct a propgrtiontests and a two gample (4 pts) orrection b) Clearly state the null and alternative hypotheses for the test. H0:p1=p2HA:p1>P2\begin{array}{l} H_{0}: p_{1}=p_{2} \\ H_{A}: p_{1}>P_{2} \end{array} (4 pts) c) Based on the technology output given below will you reject the null hypothesis? Explain why or why not? (4 pis)
Hypothesis test results: \begin{tabular}{|l|r|r|r|r|c|c|c|c|} \hline Difference & Count1 & Total1 & Count2 & Total2 & Sample Diff & Std. Err. & Z-Stat & P-value \\ \hline P1P2\mathrm{P}_{1}-\mathrm{P}_{2} & 548 & 978 & 477 & 915 & 0.039015723 & 0.022917462 & 1.7024452 & 0.0443 \\ \hline \end{tabular}
We fiwie reject the null hypothgsis Becaus e there enough why? - significant evidence why d) Write a sentence clearly stating your conclusion in the context of the problem. (5 pts)

Studdy Solution

STEP 1

1. We are comparing proportions of satisfaction with the quality of the environment between two years: 2016 and 2018.
2. The significance level for the hypothesis test is α=0.05 \alpha = 0.05 .

STEP 2

1. Determine the type of hypothesis test.
2. State the null and alternative hypotheses.
3. Analyze the hypothesis test results.
4. Draw a conclusion based on the test results.

STEP 3

We will conduct a hypothesis test for proportions, not means, because we are comparing the proportion of satisfied individuals in two different years. This is a two-sample test because we have two independent samples: one from 2016 and one from 2018.

STEP 4

State the null and alternative hypotheses:
H0:p1=p2(The proportion of satisfied individuals is the same in 2016 and 2018)H_{0}: p_{1} = p_{2} \quad \text{(The proportion of satisfied individuals is the same in 2016 and 2018)}
HA:p1>p2(The proportion of satisfied individuals is greater in 2016 than in 2018)H_{A}: p_{1} > p_{2} \quad \text{(The proportion of satisfied individuals is greater in 2016 than in 2018)}

STEP 5

Analyze the hypothesis test results:
- Sample difference (P1P2 \mathrm{P}_{1} - \mathrm{P}_{2} ) is 0.039015723 0.039015723 . - Standard error is 0.022917462 0.022917462 . - Z-Statistic is 1.7024452 1.7024452 . - P-value is 0.0443 0.0443 .
Since the P-value 0.0443 0.0443 is less than the significance level α=0.05 \alpha = 0.05 , we reject the null hypothesis.

STEP 6

Conclusion:
Based on the hypothesis test, we reject the null hypothesis. There is significant evidence at the 0.05 0.05 significance level to conclude that the proportion of Americans satisfied with the quality of the environment has declined from 2016 to 2018.

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